The shaded region \(ABC\) shown in the diagram is given by the inequalities
Show Hint
In graph-based inequality questions, first identify the boundary lines, then test a point from the shaded region to decide whether the sign is \(\leq\) or \(\geq\).
Step 1: Understanding the Concept:
The problem asks to identify the set of linear inequalities that defines the shaded triangular region shown in the diagram. The vertices of the triangle are given as A(0,3), B(3,0), and C(5,0). We need to find the equations of the lines forming the sides of the triangle and then determine the correct inequality sign for each by considering the location of the shaded region. Step 2: Key Formula or Approach:
1. Find the equation of the line passing through points A(0,3) and B(3,0).
2. Find the equation of the line passing through points A(0,3) and C(5,0).
3. The third side of the triangle lies on the x-axis, which is given by the equation \( y=0 \).
4. Determine the inequality for each line. For a line \( ax+by=c \), the region on one side is \( ax+by \le c \) and on the other is \( ax+by \ge c \). We can test a point inside the shaded region (e.g., a point on the line segment BC like (4,0) is not inside, let's try (3, 0.5)) to find the correct direction of the inequality. Step 3: Detailed Explanation:
The shaded region is in the first quadrant, so we know \( x \ge 0 \) and \( y \ge 0 \). Line 1: Passing through A(0,3) and B(3,0)
Using the intercept form \( \frac{x}{a} + \frac{y}{b} = 1 \), where \( a=3 \) and \( b=3 \):
\[ \frac{x}{3} + \frac{y}{3} = 1 \implies x + y = 3 \]
The shaded region (triangle ABC) is above this line. Let's test a point in the region, for example, the vertex C(5,0) which lies on the boundary of the region.
For point C(5,0): \( 5+0 = 5 \). Since \( 5 \ge 3 \), the inequality for this boundary is \( x+y \ge 3 \). Line 2: Passing through A(0,3) and C(5,0)
Using the intercept form, where \( a=5 \) and \( b=3 \):
\[ \frac{x}{5} + \frac{y}{3} = 1 \]
Multiplying by the LCM (15) to clear the denominators:
\[ 3x + 5y = 15 \]
The shaded region is below this line. Let's test a point in the region, for example, the vertex B(3,0).
For point B(3,0): \( 3(3) + 5(0) = 9 \). Since \( 9 \le 15 \), the inequality for this boundary is \( 3x + 5y \le 15 \). Line 3: The base of the triangle lies on the x-axis (segment BC)
The equation for the x-axis is \( y = 0 \). Since the entire region is above the x-axis, the inequality is \( y \ge 0 \), which is already part of the standard constraints.
Combining the inequalities, we get:
\( x+y \ge 3 \)
\( 3x+5y \le 15 \)
\( x \ge 0, y \ge 0 \) Step 4: Final Answer:
The set of inequalities describing the shaded region is \( x + y \ge 3, 3x + 5y \le 15, x \ge 0, y \ge 0 \). This corresponds to option (B).
